We let \({K}\)  denote the category of \(k\) -spaces [1]}
and we let \({U}\)  denote the full subcategory of compactly generated spaces (i.e. weakly Hausdorff \(k\) -spaces).
Let \(B\)  be an object of \({U}\) . We will be interested in the category \({K}/B\)  of spaces over \(B\) .
Recall from [2]} that \({K}/B\)  is a topological bicomplete category,
in the sense that \({K}/B\)  is enriched over \({K}\) , the underlying category is complete and
cocomplete, and that it is tensored and cotensored over
\({K}\) . For any space \(K\)  and space \(X\)  over \(B\)  the tensor \(K\otimes X\)  is
defined to be the space \(K\times X\)  in \({K}\) , considered as a space over \(B\)  via the obvious map \(K\times X\rightarrow B\) .
Similarly, the cotensor \(X^K\)  is defined to be the space \(\mathrm {Map}_B(K,X)\)  given by the pullback diagram
\({\mathrm {Map}_B(K,X) [r] [d] & \mathrm {Map}(K,X) [d] \\B [r] & \mathrm {Map}(K,B) }\) 
